# hw01 - Q × with division d The set P of 2 × 2 matrices...

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Modern Algebra 1 Homework 1.1 Spring 2010 Due January 20 Exercise 1. Let G be a nonempty set with a binary operation. We say G is a monoid if: (i) a ( bc ) = ( ab ) c for all a,b,c G and (ii) there is an e G (called an identity ) so that ae = ea = a for all a G . Note that in particular every group is a monoid. Which of the following sets with binary operations are monoids? Be sure to justify your answers! a. N with multiplication. b. N with addition. c.
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Unformatted text preview: Q × with division. d. The set P of 2 × 2 matrices with positive real entries, and matrix multiplication. Exercise 2. Prove that the identity element in any monoid (and hence any group) is unique. [ Hint: If you have two identities, what happens when you multiply them together?] Exercise 3. Prove that matrix multiplication is an associative operation on M 2 ( R )....
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